Rigid formation control of double-integrator systems

ABSTRACT

In this paper, we study rigid formation control systems modelled by double integrators. Two kinds of double-integrator formation systems are considered, namely formation stabilisation systems and flocking control systems. Novel observations on the measurement requirement, the null space and eigenvalues of the system Jacobian matrix will be provided, which reveal important properties of system dynamics and the associated convergence results. We also establish some new links between single-integrator formation systems and double-integrator formation systems via a parameterised Hamiltonian system, which, in addition, provide novel stability criteria for different equilibria in double-integrator formation systems by using available results in single-integrator formation systems.

KEYWORDS:

• Rigid formation
• double-integrator model
• parameterised Hamiltonian system
• equilibrium analysis

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. The realisation of a target formation with the given desired distances may not be unique up to rotation and translation (Hendrickson, 1992).

2. Note that there are two types of velocity consensus algorithms depending on different underlying graphs: one is based on undirected underlying graph and the other is based on directed graph (for achieving a leader-following control). Furthermore, the underlying graph for achieving velocity consensus can be different to the one of shape stabilisation (see relevant discussions in Deghat et al., 2016; Qin & Yu, 2013). In this paper, we focus on the first one (with undirected underlying graph for the velocity consensus) and assume the same underlying graph for both shape stabilisation and velocity consensus.

3. For example, a target triangle formation shape with three given distances satisfying triangle inequality can be realised in ${\mathbb{R}}^2$ but not in ${\mathbb{R}}^1$. In this case, any collinear equilibrium that only spans d = 1-dimensional affine space is a degenerate equilibrium.

4. Two flows ϕ_t: A → A and ψ_t: B → B are conjugate if there exists a homeomorphism h: A → B such that for each x ∈ A and $t\in \mathbb{R}$, there holds h(ϕ_t(x)) = ψ_t(h(x)). For more discussions on topological conjugacy, the readers are referred to Meiss (2007, Chapter 4.7).

5. Another way for proving the exponential convergence is to focus on the relative position dynamics or distance error dynamics (see e.g. Dörfler & Francis, 2010; Sun, Mou, Anderson, et al., 2016).

Additional information

Funding

This work is supported by NICTA, which is funded by the Australian Government as represented by the Department of Broadband, Communications and the Digital Economy and the Australian Research Council through the ICT Centre of Excellence program. It is also partially supported by National Natural Science Foundation of China [grant number 61501282]. B. D. O. Anderson was supported by the ARC [grant number DP130103610]. H.-S. Ahn is supported by the National Research Foundation of Korea [grant number NRF-2013R1A2A2A01067449]. Z. Sun is supported by the Prime Minister's Australia Asia Incoming Endeavour Postgraduate Award from Australian Government.